A Revolutionary Material
نویسنده
چکیده
This is an expository introduction, for a general mathematics audience, to the modeling of the fluid/solid phase transition and in particular to complications created by the discovery of quasicrystals. One goal is to elucidate certain features of the modeling which are ripe for mathematical investigation. * Research supported in part by NSF Grant DMS-1208941 The 2011 Nobel prize for chemistry was awarded to Dan Schectmann for the discovery of quasicrystals, an exotic class of materials. The discovery was published in 1984 and was quickly treated as revolutionary, with front page headlines in newspapers. While the award was for chemistry, the revolution was more broadly based within the interdisciplinary subject of materials science. This can be described easily and we will begin with a sketch of this. The multifaceted implications for mathematics are more complicated and we will try to elucidate them afterwards. The basic fact is that quasicrystals are equilibrium solids which are not crystalline. Not only is their pattern of atoms not crystalline, the pattern has a fascinating hierarchical structure. However we emphasize that the hierarchical pattern is not essential to the revolutionary significance of quasicrystals to materials science. It had been understood for many years, following the development of X-ray diffraction, that common inorganic solids (for instance all solids composed of only one chemical element) are crystalline, and great practical success followed from incorporating this in their modeling, essentially by analyzing various perturbations of a crystalline atomic configuration. This is evident from standard textbooks on solid state physics from the 1970’s. The startling fact uncovered by the discovery of quasicrystals was the existence of a previously unknown class of inorganic solids, of unknown diversity, for which a fundamentally different approach would be needed, specifically without the help of an underlying crystalline structure. That was the revolution in materials science. As for the implications for mathematics, one path quickly developed from the hierarchical atomic patterns, which played a central role in the theory of Levine and Steinhardt based on aperiodic tilings such as the Penrose ‘kites and darts’ (see Figure 1). Figure 1. A Penrose kite and dart tiling of the plane
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